An ultra-wideband origami microwave absorber

Microwave absorbers have been used to mitigate signal interference, and to shield electromagnetic systems. Two different types of absorbers have been presented: (a) low-cost narrowband absorbers that are simple to manufacture, and (b) expensive wideband microwave absorbers that are based on complex designs. In fact, as designers try to increase the bandwidth of absorbers, they typically increase their complexity with the introduction of several electromagnetic components (e.g., introduction of multi-layer designs, introduction of multiple electromagnetic resonators, etc.,), thereby increasing their fabrication cost. Therefore, it has been a challenge to design wideband absorbers with low cost of fabrication. To address this challenge, we propose a novel design approach that combines origami math with electromagnetics to develop a simple to manufacture ultra-wideband absorber with minimal fabrication and assembly cost. Specifically, we utilize a Tachi–Miura origami pattern in a honeycomb configuration to create the first absorber that can maintain an absorptivity above 90% in a 24.6:1 bandwidth. To explain the ultra-wideband behavior of our absorber, we develop analytical models based on the transmission-reflection theory of electromagnetic waves through a series of inhomogeneous media. The ultra-wideband performance of our absorber is validated and characterized using simulations and measurements.

In this section, we theoretically explain and prove the ultra-wideband response of the proposed TMP absorber. The question that needs to be answered is: what is the main characteristic that introduces this behavior? Our hypothesis is that the ultra-wideband performance is a result of the periodic placement of dielectric and air slabs, which is not done in traditional homogeneous absorbers. To prove our assumption we will compare the achievable BW of an inhomogeneous absorber with the achievable bandwidth of a homogeneous one. In both cases, the absorbers are coated with highly resistive ink, which is represented here as a surface impedance Z ink . Also, both absorbers are backed by a perfectly conductive layer. To compare the achievable BWs, we use Rozanov's work, 1 where he showed that the operating BW of a radar absorber is bounded by the following equation: where Γ is the reflection coefficient in the operating frequency range f min to f max corresponding to wavelength range λ max to λ min , µ s,i is the static permeability (in our case is always 1), and d i is the thickness of the i th layer of the multi-layer slab. From equation (1) we see that the bandwidth corresponding to λ max to λ min is inversely proportional to the reflection coefficient. Therefore, if we show that the reflection coefficient of the inhomogeneous absorber is smaller than the reflection coefficient of the homogeneous one, correspondingly the bandwidth of the inhomogeneous absorber will be larger than the bandwidth of the homogeneous one. In what follows, we present an analytical approach based on transmission-reflection theory, 2 which proves that the inhomegenous multi-layered dielectric structures that are coated with impedance layers exhibit larger bandwidth than homogeneous dielectrics that are coated with an impedance layer.

Transmission-Reflection Theory
First, we consider the traditional case of two homogeneous domains (Domain I and II) backed by a perfect electric conducting (PEC) plate, shown in Fig. S1a. Using transmission-reflection theory 2 at the interface of the two domains, we find the electric Note that the direction of the vector over the electric field denotes the direction of propagation, which is towards the +ẑ for the right vector, and towards the −ẑ for the left vector, respectively. Here, Γ b and T b are the reflection and transmission coefficients, respectively, at the interface of the two domains. Magnetic field equations can be similarly written and they are omitted here for reasons of brevity. By applying the appropriate boundary condition, we obtain the following equations: 1

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Solving equations (7) and (8), gives us Next, we consider the case of Fig. S1b, where we define domain I as air, domain II as dielectric and an infinitesimally thin layer of impedance Z in between. By applying conditions in equation (9) and equation (10), we have: Here, the notation k, l of the the superscripts of Γ and T is used to define the direction of arrival of the impinging wave. For example, when it is k, l = I, Z, impinging wave travelling in domain I impinges on the Z impedance layer. Symbols η 0 and η denote the intrinsic impedances of domains I and II, respectively, and Z is the characteristic impedance of the impedance layer. The thickness, t, of the impedance layer is assumed to be infinitesimal (t → 0), while the thickness of domain II is d. By using equations (2)-(6) and equations (11)-(18) we can write the electric field of each domain as: Domain I: T III,Z = T I,Z

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Therefore, the corresponding electric field components in each domain are written as follows: Domain I: Domain II: Domain III: To find the reflected E-field component in domain I we use equation (46): where, By substituting equations (68)-(72) into equation (67), we can express ← − E r in terms of the incident wave component, − → E i . Consequently, the total reflection can be calculated using equation (30). Figure S2 shows the reflection coefficients for both the inhomogeneous and homogeneous absorbers. It is seen that the reflection coefficient of the inhomogeneous medium is lower than the reflection coefficient of the homogeneous medium in the entire range of frequencies. Therefore, based on the relationship (which was explained above) between the reflection coefficient and the bandwidth in equation (1), we expect that an absorber that has the same properties with the inhomogeneous medium will exhibit wider bandwidth than an absorber that has the same properties with the homogeneous medium.

Tachi-Miura Polyhedron (TMP) Absorber
An extensive analysis of the Tachi-Miura Polyhedron (TMP) can be found in literature. 3,4 For the readers' convenience, we concisely present here the TMP origami design process.

Definition of Tachi-Miura Polyhedron
The Tachi-Miura Polyhedron (TMP) is defined by defining the coordinates in each panel in each panel's local coordinate frame as shown in Fig. S3. The coordinates of each panel around the first vertex are as follows: (73) Figure S3 shows how the initial vertices were determined. The panel coordinates are then rotated to the desired position in the global coordinate frame (γ 1 = 80 o ) using the known angles between the panels (α i ), and the actuation angle (γ 1 ) where R i is the rotation matrix of the i th panel and R x and R z are the rotation matrices about x and z, respectively. Also γ 1 γ 4 can be found from γ 1 using well-known relations. This process is repeated for the second vertex shown with an added displacement from the first vertex, and for the vertices on the back side of the pattern.

TMP -3D Electromagnetic CAD Model
To model and study our TMP absorber ANSYS HFSS is used. Figures S4 and S5 show the steps that we follow to construct the TMP structure in ANSYS HFSS using its 3D modeler.
Specifically, Fig. S4 shows in detail how a flat (i.e., unfolded) panel (that is the main building block of the TMP unit cell) folds to the final folded panel. Figure S5, in turn, shows all the folding steps of the TMP unit cell geometry. Specifically, first, Fig. S5a presents the unfolded (i.e., flat) origami pattern of the front panel of the TMP unit cell with its corresponding geometrical parameters of length (l and m), width (d), and internal angle (α). Second, the origami pattern is folded as shown in Fig. S5b, and the geometric 3 Consequently, if we combine equations (76) and (78), it can be found that θ g ∈ [0, 2α]. For our design we specify the values of l, m, d, and α as shown in Table T1. Then, parameter θ m is calculated by optimizing the performance of our TMP absorber in terms of bandwidth using full-wave simulations. As shown in Fig. S6 the optimal value of θ m is found to be 40 o (for this value of θ m , θ g is 99.85 o ), while for greater angles of θ m , the performance of our absorber degrades significantly. For example, in the entire bandwidth of operation, an average of 20 dB reduction in absorption is observed as θ m increases from 40 o to 80 o . Notably, the value of θ m is not changed throughout our analysis as we assume that our design operates only at one fixed state of the TMP origami pattern. The rest dependent geometric parameters of our design, B, W and H are calculated using equations (75)-(77), and they are shown in Table T1. Also, it should be pointed out that θ m cannot be less than 36 o for the specific geometry of our design, as this will cause the intermediate walls of the TMP structure to collide. Third, we complete the TMP unit cell by doing the following: (a) we mirror the folded pattern of  Table T2.

TMP -Absorptivity Analysis
As discussed in Section 2.1.3 of the main manuscript, our TMP absorber provides an ultrawideband performance with a 24.6 : 1 bandwidth for both the broadside and transverse illumination and for incident waves with angles of incidence that are less than 40 o . Specifically, our simulation results showed that for both the broadside (φ = 0 o , θ = 0 o ) and transverse (φ = 0 o , θ = 90 o ) case, and incidence waves with angles of incidence θ = 0 o and θ = 90 o , respectively, the TMP origami absorber provides for both TE and TM modes a reflection that is below 10 dB in the frequency band between 1.22 GHz and 30 GHz (see Figs. 5d and 5g in the main manuscript). In addition, Figs. 5e, 5f and 5h, 5i of the main manuscript illustrated consistent absorption of EM waves for incidence angles less than 40 o and without sacrificing bandwidth for both the TE and TM impinging waves. Notably, for incidence angles greater than 40 o , we showed that TE impinging waves are more strongly reflected, thereby, reducing the corresponding operational bandwidth of the absorber. To explain this behavior, we use the boundary conditions of electromagnetic waves. Let us take the case of an impinging TE wave as shown in Fig. S7a. For reasons of simplicity and without loss of generality, we make the following assumptions: a) the electric field component of the incident wave, ⃗ E inc , is parallel to theŷ-direction (the local x ′ y ′ z ′ coordinate system shown in Fig. S7a is parallel to the global xyz coordinate system), b) we model a plane wave and, therefore, the electric ( ⃗ E inc ) and magnetic ( ⃗ H inc ) field components are always perpendicular to each other and they are also perpendicular to the direction of propagationk, and c) the incident wave (see the direction of propagationk of the incident wave in Fig. S7a) belongs to the cut plane φ = 0 o and forms an angle θ with axis-z; notably, all our studies in Section 2.1.3 of the main manuscript follow the same assumptions. Figure S7b shows equivalently the case of a TM incident wave, where now the magnetic field component ( ⃗ H inc ) is taken to be parallel to theŷ-direction. What is important to note here is the orientation of the magnetic field component for both the TE and TM impinging waves, as the angle of incidence θ changes. Namely, as we can see from Fig. S7a, as the angle θ increases from 0 o to 90 o , the magnetic field component of the incident wave changes its orientation from solely parallel to nearly perpendicular in respect to the absorber's aperture. On the other hand, in the TM impinging wave case (see Fig. S7b) the corresponding magnetic field component remains parallel for any angle of incidence, θ . Next, let a plane wave with angle of incidence θ impinge on a perfect electric conductive (PEC) plane defined by its unit normal vectorn. The boundary conditions are, 5 : , ρ s and ⃗ J s are the electric surface charge density and the electric current density, respectively, and µ and ε are the magnetic permeability and electric permittivity of the medium, respectively. Equation (79) shows that, any magnetic field that is tangential to a conductive surface excites equivalent electric currents that flow on this surface. If the surface is purely conductive (e.g., PEC), these currents radiate back the entire energy of the incident field and this is indeed what we observe when we have a perfect conductor. However, when the surface is resistive, like the absorber in our case, these currents are absorbed and no radiation occurs. On the other hand, as we can see from (81), any magnetic field that is perpendicular to a conductive surface is always reflected back;n · µ ⃗ H total = 0 → ⃗ H incident = − ⃗ H scattered . This is exactly what we see with the TE impinging waves. As the angle of incidence θ increases, the magnetic field component becomes closer to be perpendicular to the aperture of the absorber, resulting into strong reflection. Figures S7c and S7d show the simulated reflectivity of the TMP absorber for different angles of incidence for the case of broadside illumination for both the TE and TM modes. To also, prove that weak or strong equivalent electric currents are excited on the aperture of our absorber due to the impinging TE and TM waves as the angle of incidence changes, we choose to plot the electric current density of both types of waves (TE and TM) at a specific frequency of operation and at two different angles of incidence where the reflectivity (or equivalently the absorptivity) changes. Namely, for a TE impinging wave, as we can see from Figs. S7e and S7g, the electric current that flows on the aperture of the absorber due to an impinging wave with angle of incidence (φ = 0 o , θ = 30 o ), is much stronger compared to the electric current that flows on the aperture of the absorber due to an impinging wave with angle of incidence (φ = 0 o , θ = 60 o ); notably, the reflectivity in the first case (see Fig. S7c) is lower than the reflectivity of the latter case. Equivalently, for all the TM mode responses, the reflectivity is low because, strong electric currents flow on the aperture of our absorber (see Figs. S7f and S7h).

Comparison of TMP to Hexagonal Honeycomb Arrays
The TMP and hexagonal honeycomb are analyzed for three different materials using ANSYS Workbench 2021 R2 Mechanical. The properties of these materials are shown in Table T3. Figure S8 shows the hexagonal honeycomb keeping the same xand y-dimensions as in the TMP, where A regular hexagon does not exactly match the aspect ratio of the selected TMP. Figure S8 illustrates this mismatch of aspect ratio. The TMP array with dimensions of the rectangular area that encompasses it (B and W ) is shown in Fig. S8a. Figure S8b shows the hexagonal honeycomb array with 3 full hexagons (equivalent to TMP array shown in Fig. S8b) spanning the W direction overflows outside the constant rectangular area in the B direction. Figure S8c shows the hexagonal honeycomb array with 3 full hexagons spanning the B direction (rotated 90 degrees and scalled) overflows outside the constant rectangular area in the W direction. All the associated parameters are given in Table T4 for both TMP and Hexagonal Honeycomb.